\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  51.0 & -6.00 x_{1} & -10.00 x_{2} & +  7.00 x_{3} & +  8.00 x_{4} & -2.00 x_{5} & -7.00 x_{6} & -8.00 x_{7}\\
 x_{9}   &  96.0 & -7.00 x_{1} & -10.00 x_{2} & -10.00 x_{3} & -6.00 x_{4} & -10.00 x_{5} & +  6.00 x_{6} & +  9.00 x_{7}\\
 x_{10}   &  81.0 & -6.00 x_{1} & -9.00 x_{2} &   & + 10.00 x_{4} & -5.00 x_{5} & -6.00 x_{6} & -9.00 x_{7}\\
 x_{11}   &  6.0 & +  2.00 x_{1} & -9.00 x_{2} & +  5.00 x_{3} & + 10.00 x_{4} & +  5.00 x_{5} &   & -6.00 x_{7}\\
 x_{12}   &  37.0 & -6.00 x_{1} & -2.00 x_{2} & -3.00 x_{3} & -1.00 x_{4} & +  2.00 x_{5} & +  9.00 x_{6} & -4.00 x_{7}\\
 x_{13}   &  17.0 & -3.00 x_{1} & +  9.00 x_{2} & -4.00 x_{3} & -6.00 x_{4} & -7.00 x_{5} &   & +  1.00 x_{7}\\
 x_{14}   &  -2.0 & -6.00 x_{1} & +  3.00 x_{2} & +  5.00 x_{3} & +  2.00 x_{4} & +  2.00 x_{5} & +  7.00 x_{6} & +  9.00 x_{7}\\
\hline
z    &  0.0 & +  1.00 x_{1} & +  2.00 x_{2} & -4.00 x_{3} & +  1.00 x_{4} & -1.00 x_{5} & +  3.00 x_{6} & -3.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  6.00 y_{8} & +  7.00 y_{9} & +  6.00 y_{10} & -2.00 y_{11} & +  6.00 y_{12} & +  3.00 y_{13} & +  6.00 y_{14}\\
 y_{2}   &  1.0 & + 10.00 y_{8} & + 10.00 y_{9} & +  9.00 y_{10} & +  9.00 y_{11} & +  2.00 y_{12} & -9.00 y_{13} & -3.00 y_{14}\\
 y_{3}   &  1.0 & -7.00 y_{8} & + 10.00 y_{9} &   & -5.00 y_{11} & +  3.00 y_{12} & +  4.00 y_{13} & -5.00 y_{14}\\
 y_{4}   &  1.0 & -8.00 y_{8} & +  6.00 y_{9} & -10.00 y_{10} & -10.00 y_{11} & +  1.00 y_{12} & +  6.00 y_{13} & -2.00 y_{14}\\
 y_{5}   &  1.0 & +  2.00 y_{8} & + 10.00 y_{9} & +  5.00 y_{10} & -5.00 y_{11} & -2.00 y_{12} & +  7.00 y_{13} & -2.00 y_{14}\\
 y_{6}   &  1.0 & +  7.00 y_{8} & -6.00 y_{9} & +  6.00 y_{10} &   & -9.00 y_{12} &   & -7.00 y_{14}\\
 y_{7}   &  1.0 & +  8.00 y_{8} & -9.00 y_{9} & +  9.00 y_{10} & +  6.00 y_{11} & +  4.00 y_{12} & -1.00 y_{13} & -9.00 y_{14}\\
\hline
z    &  -0 & -51.00 y_{8} & -96.00 y_{9} & -81.00 y_{10} & -6.00 y_{11} & -37.00 y_{12} & -17.00 y_{13} & +  2.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.66666666667 & + 11.33 y_{8} & +  1.00 y_{9} & + 12.00 y_{10} & +  2.00 y_{11} & +  8.67 y_{12} & +  2.33 y_{13} & -0.67 y_{7}\\
 y_{2}   &  0.666666666667 & +  7.33 y_{8} & + 13.00 y_{9} & +  6.00 y_{10} & +  7.00 y_{11} & +  0.67 y_{12} & -8.67 y_{13} & +  0.33 y_{7}\\
 y_{3}   &  0.444444444444 & -11.44 y_{8} & + 15.00 y_{9} & -5.00 y_{10} & -8.33 y_{11} & +  0.78 y_{12} & +  4.56 y_{13} & +  0.56 y_{7}\\
 y_{4}   &  0.777777777778 & -9.78 y_{8} & +  8.00 y_{9} & -12.00 y_{10} & -11.33 y_{11} & +  0.11 y_{12} & +  6.22 y_{13} & +  0.22 y_{7}\\
 y_{5}   &  0.777777777778 & +  0.22 y_{8} & + 12.00 y_{9} & +  3.00 y_{10} & -6.33 y_{11} & -2.89 y_{12} & +  7.22 y_{13} & +  0.22 y_{7}\\
 y_{6}   &  0.222222222222 & +  0.78 y_{8} & +  1.00 y_{9} & -1.00 y_{10} & -4.67 y_{11} & -12.11 y_{12} & +  0.78 y_{13} & +  0.78 y_{7}\\
 y_{14}   &  0.111111111111 & +  0.89 y_{8} & -1.00 y_{9} & +  1.00 y_{10} & +  0.67 y_{11} & +  0.44 y_{12} & -0.11 y_{13} & -0.11 y_{7}\\
\hline
z    &  0.222222222222 & -49.22 y_{8} & -98.00 y_{9} & -79.00 y_{10} & -4.67 y_{11} & -36.11 y_{12} & -17.22 y_{13} & -0.22 y_{7}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  49.2222222222 & -11.33 x_{1} & -7.33 x_{2} & + 11.44 x_{3} & +  9.78 x_{4} & -0.22 x_{5} & -0.78 x_{6} & -0.89 x_{14}\\
 x_{9}   &  98.0 & -1.00 x_{1} & -13.00 x_{2} & -15.00 x_{3} & -8.00 x_{4} & -12.00 x_{5} & -1.00 x_{6} & +  1.00 x_{14}\\
 x_{10}   &  79.0 & -12.00 x_{1} & -6.00 x_{2} & +  5.00 x_{3} & + 12.00 x_{4} & -3.00 x_{5} & +  1.00 x_{6} & -1.00 x_{14}\\
 x_{11}   &  4.66666666667 & -2.00 x_{1} & -7.00 x_{2} & +  8.33 x_{3} & + 11.33 x_{4} & +  6.33 x_{5} & +  4.67 x_{6} & -0.67 x_{14}\\
 x_{12}   &  36.1111111111 & -8.67 x_{1} & -0.67 x_{2} & -0.78 x_{3} & -0.11 x_{4} & +  2.89 x_{5} & + 12.11 x_{6} & -0.44 x_{14}\\
 x_{13}   &  17.2222222222 & -2.33 x_{1} & +  8.67 x_{2} & -4.56 x_{3} & -6.22 x_{4} & -7.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
 x_{7}   &  0.222222222222 & +  0.67 x_{1} & -0.33 x_{2} & -0.56 x_{3} & -0.22 x_{4} & -0.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
\hline
z    &  -0.222222222222 & -1.67 x_{1} & -0.67 x_{2} & -0.44 x_{3} & -0.78 x_{4} & -0.78 x_{5} & -0.22 x_{6} & -0.11 x_{14}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  49.2222222222 & -11.33 x_{1} & -7.33 x_{2} & + 11.44 x_{3} & +  9.78 x_{4} & -0.22 x_{5} & -0.78 x_{6} & -0.89 x_{14}\\
 x_{9}   &  98.0 & -1.00 x_{1} & -13.00 x_{2} & -15.00 x_{3} & -8.00 x_{4} & -12.00 x_{5} & -1.00 x_{6} & +  1.00 x_{14}\\
 x_{10}   &  79.0 & -12.00 x_{1} & -6.00 x_{2} & +  5.00 x_{3} & + 12.00 x_{4} & -3.00 x_{5} & +  1.00 x_{6} & -1.00 x_{14}\\
 x_{11}   &  4.66666666667 & -2.00 x_{1} & -7.00 x_{2} & +  8.33 x_{3} & + 11.33 x_{4} & +  6.33 x_{5} & +  4.67 x_{6} & -0.67 x_{14}\\
 x_{12}   &  36.1111111111 & -8.67 x_{1} & -0.67 x_{2} & -0.78 x_{3} & -0.11 x_{4} & +  2.89 x_{5} & + 12.11 x_{6} & -0.44 x_{14}\\
 x_{13}   &  17.2222222222 & -2.33 x_{1} & +  8.67 x_{2} & -4.56 x_{3} & -6.22 x_{4} & -7.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
 x_{7}   &  0.222222222222 & +  0.67 x_{1} & -0.33 x_{2} & -0.56 x_{3} & -0.22 x_{4} & -0.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
\hline
z    &  -0.666666666667 & -1.00 x_{1} & +  3.00 x_{2} & -2.33 x_{3} & +  1.67 x_{4} & -0.33 x_{5} & +  5.33 x_{6} & -0.33 x_{14}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  49.2222222222 & -11.33 x_{1} & -7.33 x_{2} & + 11.44 x_{3} & +  9.78 x_{4} & -0.22 x_{5} & -0.78 x_{6} & -0.89 x_{14}\\
 x_{9}   &  98.0 & -1.00 x_{1} & -13.00 x_{2} & -15.00 x_{3} & -8.00 x_{4} & -12.00 x_{5} & -1.00 x_{6} & +  1.00 x_{14}\\
 x_{10}   &  79.0 & -12.00 x_{1} & -6.00 x_{2} & +  5.00 x_{3} & + 12.00 x_{4} & -3.00 x_{5} & +  1.00 x_{6} & -1.00 x_{14}\\
 x_{11}   &  4.66666666667 & -2.00 x_{1} & -7.00 x_{2} & +  8.33 x_{3} & + 11.33 x_{4} & +  6.33 x_{5} & +  4.67 x_{6} & -0.67 x_{14}\\
 x_{12}   &  36.1111111111 & -8.67 x_{1} & -0.67 x_{2} & -0.78 x_{3} & -0.11 x_{4} & +  2.89 x_{5} & + 12.11 x_{6} & -0.44 x_{14}\\
 x_{13}   &  17.2222222222 & -2.33 x_{1} & +  8.67 x_{2} & -4.56 x_{3} & -6.22 x_{4} & -7.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
 x_{7}   &  0.222222222222 & +  0.67 x_{1} & -0.33 x_{2} & -0.56 x_{3} & -0.22 x_{4} & -0.22 x_{5} & -0.78 x_{6} & +  0.11 x_{14}\\
\hline
z    &  -0.666666666667 & -1.00 x_{1} & +  3.00 x_{2} & -2.33 x_{3} & +  1.67 x_{4} & -0.33 x_{5} & +  5.33 x_{6} & -0.33 x_{14}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  44.3333333333 & -26.00 x_{1} & + 22.00 x_{7} & + 23.67 x_{3} & + 14.67 x_{4} & +  4.67 x_{5} & + 16.33 x_{6} & -3.33 x_{14}\\
 x_{9}   &  89.3333333333 & -27.00 x_{1} & + 39.00 x_{7} & +  6.67 x_{3} & +  0.67 x_{4} & -3.33 x_{5} & + 29.33 x_{6} & -3.33 x_{14}\\
 x_{10}   &  75.0 & -24.00 x_{1} & + 18.00 x_{7} & + 15.00 x_{3} & + 16.00 x_{4} & +  1.00 x_{5} & + 15.00 x_{6} & -3.00 x_{14}\\
 x_{11}   &  8.881784197e-16 & -16.00 x_{1} & + 21.00 x_{7} & + 20.00 x_{3} & + 16.00 x_{4} & + 11.00 x_{5} & + 21.00 x_{6} & -3.00 x_{14}\\
 x_{12}   &  35.6666666667 & -10.00 x_{1} & +  2.00 x_{7} & +  0.33 x_{3} & +  0.33 x_{4} & +  3.33 x_{5} & + 13.67 x_{6} & -0.67 x_{14}\\
 x_{13}   &  23.0 & + 15.00 x_{1} & -26.00 x_{7} & -19.00 x_{3} & -12.00 x_{4} & -13.00 x_{5} & -21.00 x_{6} & +  3.00 x_{14}\\
 x_{2}   &  0.666666666667 & +  2.00 x_{1} & -3.00 x_{7} & -1.67 x_{3} & -0.67 x_{4} & -0.67 x_{5} & -2.33 x_{6} & +  0.33 x_{14}\\
\hline
z    &  1.33333333333 & +  5.00 x_{1} & -9.00 x_{7} & -7.33 x_{3} & -0.33 x_{4} & -2.33 x_{5} & -1.67 x_{6} & +  0.67 x_{14}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  44.3333333333 & +  1.62 x_{11} & -12.12 x_{7} & -8.83 x_{3} & -11.33 x_{4} & -13.21 x_{5} & -17.79 x_{6} & +  1.54 x_{14}\\
 x_{9}   &  89.3333333333 & +  1.69 x_{11} & +  3.56 x_{7} & -27.08 x_{3} & -26.33 x_{4} & -21.90 x_{5} & -6.10 x_{6} & +  1.73 x_{14}\\
 x_{10}   &  75.0 & +  1.50 x_{11} & -13.50 x_{7} & -15.00 x_{3} & -8.00 x_{4} & -15.50 x_{5} & -16.50 x_{6} & +  1.50 x_{14}\\
 x_{1}   &  5.55111512313e-17 & -0.06 x_{11} & +  1.31 x_{7} & +  1.25 x_{3} & +  1.00 x_{4} & +  0.69 x_{5} & +  1.31 x_{6} & -0.19 x_{14}\\
 x_{12}   &  35.6666666667 & +  0.62 x_{11} & -11.12 x_{7} & -12.17 x_{3} & -9.67 x_{4} & -3.54 x_{5} & +  0.54 x_{6} & +  1.21 x_{14}\\
 x_{13}   &  23.0 & -0.94 x_{11} & -6.31 x_{7} & -0.25 x_{3} & +  3.00 x_{4} & -2.69 x_{5} & -1.31 x_{6} & +  0.19 x_{14}\\
 x_{2}   &  0.666666666667 & -0.12 x_{11} & -0.38 x_{7} & +  0.83 x_{3} & +  1.33 x_{4} & +  0.71 x_{5} & +  0.29 x_{6} & -0.04 x_{14}\\
\hline
z    &  1.33333333333 & -0.31 x_{11} & -2.44 x_{7} & -1.08 x_{3} & +  4.67 x_{4} & +  1.10 x_{5} & +  4.90 x_{6} & -0.27 x_{14}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  5.88607594937 & +  0.90 x_{11} & -13.66 x_{7} & +  2.82 x_{3} & +  0.43 x_{9} & -3.78 x_{5} & -15.16 x_{6} & +  0.80 x_{14}\\
 x_{4}   &  3.39240506329 & +  0.06 x_{11} & +  0.14 x_{7} & -1.03 x_{3} & -0.04 x_{9} & -0.83 x_{5} & -0.23 x_{6} & +  0.07 x_{14}\\
 x_{10}   &  47.8607594937 & +  0.99 x_{11} & -14.58 x_{7} & -6.77 x_{3} & +  0.30 x_{9} & -8.85 x_{5} & -14.65 x_{6} & +  0.97 x_{14}\\
 x_{1}   &  3.39240506329 & +  0.00 x_{11} & +  1.45 x_{7} & +  0.22 x_{3} & -0.04 x_{9} & -0.14 x_{5} & +  1.08 x_{6} & -0.12 x_{14}\\
 x_{12}   &  2.87341772152 & +  0.01 x_{11} & -12.43 x_{7} & -2.22 x_{3} & +  0.37 x_{9} & +  4.50 x_{5} & +  2.78 x_{6} & +  0.57 x_{14}\\
 x_{13}   &  33.1772151899 & -0.75 x_{11} & -5.91 x_{7} & -3.34 x_{3} & -0.11 x_{9} & -5.18 x_{5} & -2.01 x_{6} & +  0.38 x_{14}\\
 x_{2}   &  5.18987341772 & -0.04 x_{11} & -0.19 x_{7} & -0.54 x_{3} & -0.05 x_{9} & -0.40 x_{5} & -0.02 x_{6} & +  0.05 x_{14}\\
\hline
z    &  17.164556962 & -0.01 x_{11} & -1.81 x_{7} & -5.88 x_{3} & -0.18 x_{9} & -2.78 x_{5} & +  3.81 x_{6} & +  0.04 x_{14}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.388146911519 & +  0.06 x_{11} & -0.90 x_{7} & +  0.19 x_{3} & +  0.03 x_{9} & -0.25 x_{5} & -0.07 x_{8} & +  0.05 x_{14}\\
 x_{4}   &  3.30243113523 & +  0.05 x_{11} & +  0.34 x_{7} & -1.07 x_{3} & -0.04 x_{9} & -0.77 x_{5} & +  0.02 x_{8} & +  0.05 x_{14}\\
 x_{10}   &  42.1761268781 & +  0.12 x_{11} & -1.39 x_{7} & -9.50 x_{3} & -0.11 x_{9} & -5.19 x_{5} & +  0.97 x_{8} & +  0.20 x_{14}\\
 x_{1}   &  3.81187395659 & +  0.07 x_{11} & +  0.47 x_{7} & +  0.42 x_{3} & -0.01 x_{9} & -0.41 x_{5} & -0.07 x_{8} & -0.07 x_{14}\\
 x_{12}   &  3.95341193656 & +  0.17 x_{11} & -14.94 x_{7} & -1.71 x_{3} & +  0.45 x_{9} & +  3.80 x_{5} & -0.18 x_{8} & +  0.72 x_{14}\\
 x_{13}   &  32.3978505843 & -0.86 x_{11} & -4.10 x_{7} & -3.71 x_{3} & -0.17 x_{9} & -4.68 x_{5} & +  0.13 x_{8} & +  0.28 x_{14}\\
 x_{2}   &  5.18311769616 & -0.04 x_{11} & -0.18 x_{7} & -0.54 x_{3} & -0.05 x_{9} & -0.40 x_{5} & +  0.00 x_{8} & +  0.04 x_{14}\\
\hline
z    &  18.6449812187 & +  0.21 x_{11} & -5.24 x_{7} & -5.17 x_{3} & -0.07 x_{9} & -3.73 x_{5} & -0.25 x_{8} & +  0.24 x_{14}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  2.6098032114 & -0.07 x_{13} & -1.18 x_{7} & -0.07 x_{3} & +  0.02 x_{9} & -0.57 x_{5} & -0.06 x_{8} & +  0.07 x_{14}\\
 x_{4}   &  5.1896655801 & -0.06 x_{13} & +  0.11 x_{7} & -1.29 x_{3} & -0.05 x_{9} & -1.05 x_{5} & +  0.02 x_{8} & +  0.07 x_{14}\\
 x_{10}   &  46.6507304117 & -0.14 x_{13} & -1.96 x_{7} & -10.01 x_{3} & -0.14 x_{9} & -5.84 x_{5} & +  0.98 x_{8} & +  0.24 x_{14}\\
 x_{1}   &  6.27212362671 & -0.08 x_{13} & +  0.16 x_{7} & +  0.14 x_{3} & -0.02 x_{9} & -0.77 x_{5} & -0.06 x_{8} & -0.04 x_{14}\\
 x_{12}   &  10.3426294821 & -0.20 x_{13} & -15.75 x_{7} & -2.44 x_{3} & +  0.41 x_{9} & +  2.88 x_{5} & -0.16 x_{8} & +  0.77 x_{14}\\
 x_{11}   &  37.4865386937 & -1.16 x_{13} & -4.74 x_{7} & -4.29 x_{3} & -0.20 x_{9} & -5.42 x_{5} & +  0.15 x_{8} & +  0.32 x_{14}\\
 x_{2}   &  3.66159604008 & +  0.05 x_{13} & +  0.01 x_{7} & -0.37 x_{3} & -0.04 x_{9} & -0.18 x_{5} & -0.01 x_{8} & +  0.03 x_{14}\\
\hline
z    &  26.6143909212 & -0.25 x_{13} & -6.25 x_{7} & -6.09 x_{3} & -0.11 x_{9} & -4.88 x_{5} & -0.22 x_{8} & +  0.30 x_{14}\\
\end{array}\]


 $ x_{14} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  12.8732782369 & -0.19 x_{13} & -0.91 x_{7} & +  0.16 x_{3} & -0.02 x_{9} & -1.83 x_{5} & -0.16 x_{8} & -1.64 x_{1}\\
 x_{4}   &  15.1680440771 & -0.18 x_{13} & +  0.37 x_{7} & -1.06 x_{3} & -0.09 x_{9} & -2.27 x_{5} & -0.07 x_{8} & -1.59 x_{1}\\
 x_{10}   &  81.4325068871 & -0.56 x_{13} & -1.05 x_{7} & -9.23 x_{3} & -0.25 x_{9} & -10.10 x_{5} & +  0.64 x_{8} & -5.55 x_{1}\\
 x_{14}   &  143.1184573 & -1.73 x_{13} & +  3.72 x_{7} & +  3.22 x_{3} & -0.46 x_{9} & -17.55 x_{5} & -1.40 x_{8} & -22.82 x_{1}\\
 x_{12}   &  121.245179063 & -1.54 x_{13} & -12.86 x_{7} & +  0.06 x_{3} & +  0.05 x_{9} & -10.72 x_{5} & -1.24 x_{8} & -17.68 x_{1}\\
 x_{11}   &  83.6721763085 & -1.72 x_{13} & -3.54 x_{7} & -3.25 x_{3} & -0.35 x_{9} & -11.08 x_{5} & -0.30 x_{8} & -7.36 x_{1}\\
 x_{2}   &  8.22314049587 & -0.01 x_{13} & +  0.13 x_{7} & -0.26 x_{3} & -0.06 x_{9} & -0.74 x_{5} & -0.05 x_{8} & -0.73 x_{1}\\
\hline
z    &  70.2341597796 & -0.77 x_{13} & -5.11 x_{7} & -5.10 x_{3} & -0.25 x_{9} & -10.23 x_{5} & -0.64 x_{8} & -6.95 x_{1}\\
\end{array}\]
Final Dictionary
Solution:  70.2341597796
Num Pivots:  7
\end{document}
